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Zbl 1200.55007
Bunke, Ulrich; Kreck, Matthias; Schick, Thomas
A geometric description of differential cohomology. (English)
[J] Ann. Math. Blaise Pascal 17, No. 1, 1-16 (2010). ISSN 1259-1734

This paper gives a geometric description of differential integral homology. A differential extension of ordinary integral cohomology $H^*$ is a functor $X \mapsto \widehat{H}^*(X)$ from the category of smooth manifolds to the category of ${\mathbb Z}$-graded groups together with natural transformations $$\align R : \widehat{H}^*(X) &\rightarrow \Omega_{cl}^*(X),\\ I : \widehat{H}^*(X) &\rightarrow H^*(X),\\ a:\Omega^{*-1}(X)/im(d) &\rightarrow \widehat{H}^*(X) \endalign$$ where $\Omega$ denotes differential forms. The authors show how the theory $\widehat{H}^*(X)$ can be constructed by using stratifolds (essentially stratified manifolds) as ``cycles'' and cobordism to give a geometric description of the theory. That this can be done depends on the fact that Stokes' theorem applies in this situation.
[Jonathan Hodgson (Philadelphia)]

MSC 2000:: *55N20 Generalized homology and cohomology theories
57R19 Algebraic topology on manifolds

Keywords: stratifold; Stokes' theorem; integration along the fiber

Cited in: Zbl 1202.19007

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